Daniel Conus

Asymptotically Optimum Distributed Estimation in the Presence of Attacks

Distributed estimation of a deterministic mean-shift parameter in additive zero-mean noise is studied when using quantized data in the presence of Byzantine attacks. Several subsets of sensors are assumed to be tampered with by adversaries using different attacks such that the compromised sensors transmit fictitious data. First, we consider the task of identifying and categorizing the attacked sensors into different groups according to distinct types of attacks. It is shown that increasing the number K of time samples at each sensor and enlarging the size N of the sensor network can both ameliorate the identification and categorization, but to different extents. As K→∞, the attacked sensors can be perfectly identified and categorized, while with finite but sufficiently large K, as N→∞, it can be shown that the fusion center can also ascertain the number of attacks and obtain an approximate categorization with a sufficiently small percentage of sensors that are misclassified. Next, in order to improve the estimation performance by utilizing the attacked observations, we consider joint estimation of the statistical description of the attacks and the parameter to be estimated after the sensors have been well categorized. When using the same quantization approach successfully employed without attacks, it can be shown that the corresponding Fisher Information Matrix (FIM) is singular. To overcome this, a time-variant quantization approach is proposed, which will provide a nonsingular FIM, provided that K ≥ 2. Furthermore, the FIM is employed to provide necessary and sufficient conditions under which utilizing the compromised sensors in the proposed fashion will lead to better estimation performance when compared to approaches where the compromised sensors are ignored.

Intermittency for the wave and heat equations with fractional noise in time

In this article, we consider the stochastic wave and heat equations driven by a Gaussian noise which is spatially homogeneous and behaves in time like a fractional Brownian motion with Hurst index H>1/2. The solutions of these equations are interpreted in the Skorohod sense. Using Malliavin calculus techniques, we obtain an upper bound for the moments of order p≥2 of the solution. In the case of the wave equation, we derive a Feynman–Kac-type formula for the second moment of the solution, based on the points of a planar Poisson process. This is an extension of the formula given by Dalang, Mueller and Tribe [Trans. Amer. Math. Soc. 360 (2008) 4681–4703], in the case H=1/2, and allows us to obtain a lower bound for the second moment of the solution. These results suggest that the moments of the solution grow much faster in the case of the fractional noise in time than in the case of the white noise in time.

Initial measures for the stochastic heat equation

We consider a family of nonlinear stochastic heat equations of the form @tu = Lu + (u) _ W , where _ W denotes space-time white noise, L the generator of a symmetric L evy process on R, and is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-eld solution for every nite initial measure u0. Tight a priori bounds on the moments of the solution are also obtained. In the particular case that Lf = cf 00 for some c > 0, we prove that if u0 is a nite measure of compact support, then the solution is with probability one a bounded function for all times t > 0.

On the chaotic character of the stochastic heat equation, before the onset of intermitttency

We consider a nonlinear stochastic heat equation @tu = 1 @xxu + (u)@xtW , where @xtW denotes space-time white noise and : R ! R is Lipschitz continuous. We establish that, at every xed time t > 0, the global behavior of the solution depends in a critical manner on the structure of the initial function u0: Under suitable technical conditions on u0 and , supx2Rut(x) is a.s. nite when u0 has compact support, whereas with probability one, lim supjxj!1ut(x)=(logjxj) 1=6 > 0 when u0 is bounded uniformly away from zero. The mentioned sensitivity to the initial data of the stochastic heat equation is a way to state that the solution to the stochastic heat equation is chaotic at xed times, well before the onset of intermittency.

Intermittency and Chaos for a Nonlinear Stochastic Wave Equation in Dimension 1

Consider a nonlinear stochastic wave equation driven by space-time white noise in dimension one. We discuss the intermittency of the solution, and then use those intermittency results in order to demonstrate that in many cases the solution is chaotic. For the most part, the novel portion of our work is about the two cases where (1) the initial conditions have compact support, where the global maximum of the solution remains bounded, and (2) the initial conditions are positive constants, where the global maximum is almost surely infinite. Bounds are also provided on the behavior of the global maximum of the solution in Case (2).